Learning intention
We are learning to work out a probability from the sample space, and to say what the numerator and denominator each mean.
Success criteria
- I can count all the possible outcomes and all the favourable ones.
- I can write the probability as a fraction, then convert it to a decimal and a percentage.
- I can explain what the top number and the bottom number of the fraction each represent.
- I can show that two spinners that look different can have the same probability.
Curriculum
AC9M6P01 — recognise that probabilities lie on numerical scales of 0–1 or 0%–100% and use estimation to assign probabilities that events occur in a given context, using common fractions, percentages and decimals.
The lesson also opens a question and deliberately refuses to answer it. Students will spin 20 times and get a result that does not match the probability they calculated. Let that sit. It is the engine of Lessons 4, 5 and 6, and answering it today wastes it.
Shape of the lesson
| Phase | Time | What happens |
|---|---|---|
| Warm-up | 5 min | Open question: an event with a 75% chance |
| Launch | 12 min | Thirteen spinners on the floor. Notice, then order them. |
| Explore | 25 min | Pairs build a spinner, calculate, then spin 20 times |
| Summarise | 8 min | Numerator, denominator, and the question we are leaving open |
| Review | 5 min | Exit ticket |
What we are learning today
We are learning to work out a probability from the sample space, and to say what the numerator and denominator each mean.
I will know I can do it when…
- I can count all the possible outcomes and all the favourable ones.
- I can write the probability as a fraction, then convert it to a decimal and a percentage.
- I can explain what the top number and the bottom number each represent.
- I can show that two spinners that look different can have the same probability.
Set your goal for today
Read the success criteria above. Which one is the biggest stretch for you? Write one thing you are going to focus on. You will come back to it at the end.
Not sure? Explaining what the denominator counts is the one most people find hardest today.
How hard is each question?
Every question in this unit is labelled so you know what kind of thinking it needs. The assessment has questions at all three levels.
SF — simple familiar
You have seen this kind of question before and it is clear what to do. One step, friendly numbers.
CF — complex familiar
You know the method, but there is more than one step, or you have to pick which method to use.
CU — complex unfamiliar
Nobody tells you the method. You work out what to do, and explain why you did it.
Warm-up · 5 minutes
Seventy-five per cent of what?
Think alone, then share with a partner, then collect four or five on the board. Push for variety: something with a spinner, something with counters, something from real life.
Listen for the student who says "three quarters" without connecting it to 75%. That is a conversion gap and the conversion mat should be on their desk from the start of Explore.
Launch · 12 minutes
Thirteen spinners on the floor
Spread the thirteen spinner cards face up on the floor with students in a circle around them. Do not hand out the worksheet yet.
Two of the thirteen. They look different. They are not.
The questions, in this order
- What do you notice? Take everything. Do not steer.
- Is the chance of landing on shaded the same for all of them?
- Which one gives you the smallest chance of shaded? The largest?
- Put them in order, smallest chance to largest. Students move the cards physically.
- Only once they are stuck: what would help you decide between these two?
Six of the pairs sit within about 0.04 of each other — 1/8 and 1/6, 1/3 and 3/8, 5/8 and 2/3, 5/6 and 7/8. You cannot order those by eye. That is the moment the class needs a number, and it should come from them rather than from you. Wait for it.
Two cards have the same probability written differently (1/4 and 2/8). Do not point this out. If a group notices, make them the experts for the Explore. If nobody notices, it comes up in the Summarise and lands harder.
Explore · 25 minutes
Build one, work it out, then spin it
Pairs. Each pair takes a blank spinner template — the templates are pre-divided into 4, 6 or 8 equal sections — and shades however many sections they choose.
Three steps, in order
- Shade it. Choose how many sections to shade. Record how many sections altogether and how many shaded.
- Work it out. Write the probability of landing on shaded as a fraction, then a decimal, then a percentage.
- Spin it. Paperclip and pencil point. Twenty spins, tally the shaded ones, write the result as a fraction out of 20.
Class spinner line
As pairs finish, they peg their spinner on a washing line across the room in order of probability. Because the templates differ, the class ends up with a genuine spread — and with at least two spinners that look different but land in the same place.
Circulating. The thing to check is not the fraction — most will get that. Check that they can say what the denominator counts. Ask: "what does the 8 tell me?" A student who says "the sections" has it; a student who says "the answer" does not.
Complexity bands on the worksheet
SF Count the sections, write the probability three ways. Sample space is visible and given.
CF Compare two spinners with different numbers of sections; find the probability of not shaded.
CU Design a spinner to a given probability, then design a different one with the same probability — and say what they have in common.
Differentiation
Support
- 4-section and 8-section templates only.
- Prompt: "How many parts altogether? How many are shaded?"
- Conversion mat on the desk from the start.
- Tally chart pre-drawn with 20 boxes.
Core
- 4, 6 and 8-section templates.
- Full worksheet including the complement item.
- Conversion mat if requested.
Extension
- Design a spinner with P(shaded) exactly 3/8. Then one where it is 0.375. Same spinner or not?
- Can you make a spinner with 8 equal sections where P(shaded) = 1/3? Explain.
- How many of the thirteen Launch spinners could be redrawn with 24 sections?
Part C · Show your thinking
A spinner has 8 equal sections. 3 sections are shaded. Write the probability of landing on shaded three ways.
The same spinner. What is the probability of landing on a section that is NOT shaded? Explain how you worked it out.
Spinner P has 4 sections with 1 shaded. Spinner Q has 8 sections with 2 shaded. Which gives a better chance of landing on shaded? How do you know?
2 5/8 = 0.625 = 62.5%. Either count the 5 unshaded sections, or do 1 − 3/8. Ask the counters whether they could have done it without counting.
3 Neither. 1/4 and 2/8 are the same number. The spinners look different and the chance is identical.
Talk it through with a partner
Show your partner your answer to question 3. Your partner picks one of these to ask you:
- How do you know they are the same and not just close?
- Could you draw a third spinner with the same chance again?
- What would change if spinner Q had 3 shaded instead of 2?
Write down the question you were asked and what you said back.
Choose your path
Pick the one that matches where you are right now. You can change your mind.
A spinner has 4 equal sections. 1 is shaded. Write the probability three ways.
The same spinner. What is the probability of NOT shaded?
A spinner has 8 sections and 4 are shaded. Write the probability three ways.
Design a spinner where the probability of shaded is exactly 3/8. Now design a different one where it is 0.375. Is it the same spinner?
Can you make a spinner with 8 equal sections where the probability of shaded is exactly 1/3? Explain your answer.
★2 No. A third of 8 is 2.67 sections, and you cannot shade part of a section. It only works when the number of sections is a multiple of 3.
Summarise · 8 minutes
Where the number comes from
Three things to land
- The denominator counts every possible outcome. The numerator counts the ones you want. Say it with a spinner in your hand, pointing at the sections.
- Spinners that look different can have the same probability. Hold up the 4-section-with-1-shaded and the 8-section-with-2-shaded. 1/4 and 2/8 are the same number.
- The probability comes from the sample space, not from what happened when you spun. This is the sentence to write on the board and keep.
Collect two or three theories. Do not settle it. Write the best version of the question on butcher's paper and leave it on the wall until Lesson 4.
The theories you will get are usually: not enough spins, the paperclip is dodgy, bad luck, or the spinner is not really equal. Three of those four are worth revisiting in Lesson 6, so write them down verbatim with the students' names next to them.
Vocabulary
Add to the Frayer boards: sample space. Complete this one together — it is the most abstract word in the cycle and the definition students write today will be reused in Lesson 7 and in assessment item A1.
Review · 5 minutes
Exit ticket
What is the probability of landing on shaded? Write it as a fraction, a decimal and a percentage.
Maya says "the probability of shaded is 7 out of 10, because that is what I spun." Is Maya right? Explain your answer.
Draw a spinner with a DIFFERENT number of sections that has the same probability of shaded as Maya’s.
Answers.
1. 4/10 = 0.4 = 40%. Accept 2/5. A student who simplifies to 2/5 and still gets 0.4 has the stronger grasp; one who writes 2/5 = 0.25 has simplified correctly and then converted the wrong fraction.
2. No. 7 out of 10 is what Maya observed when she spun it. The probability is 4/10, and it comes from the spinner itself — the sections did not change because of what happened when she spun. Maya's result is evidence, not the probability.
3. Any equivalent with a different total: 2 of 5, 6 of 15, 8 of 20. A student who draws 4 of 10 again has answered a different question. Support version: 5 of 10 = 1/2, so 1 of 2, 2 of 4, 3 of 6 all work.
What to sort for. Question 2 is the discriminator, and it matters more than question 3. A student who agrees with Maya has swapped observed frequency for probability — the same confusion Sam carries in the Lesson 4 exit ticket, and the one that makes Lesson 6 impossible. Note those students now; they will need reaching twice.
Question 3 sorts for readiness: students producing 2 of 5 or 8 of 20 have grasped that the sample space can be resized without changing the probability, and are ready for the extending prompts in Lesson 3.
Think back
Your goal
Go back to the Intentions tab and read the goal you set. Did you meet it?
Nothing you type here is saved. Copy it into your book before you close the page.
What you need
- resources/l2-01-worksheet-core.pdf — 3 pages, most of the class
- resources/l2-02-worksheet-support.pdf — 3 pages, for students grouped S from the pre-test
- resources/l2-03-spinner-cards.pdf — 13 cards, two per A4 landscape. Print once on card and laminate; they are used again in Lesson 3. The teacher key is the last page — keep it.
- resources/l2-04-blank-spinners.pdf — 4, 6 and 8-section templates, four per page. One per pair.
- shared/conversion-mat.pdf — shared resource
- shared/frayer-boards.pdf — print the sample space page
Equipment
- Paperclips — one per pair, for spinning
- Pencils, colouring materials for shading
- Washing line and pegs across the room for the class spinner line
- Butcher's paper for the open question, to stay up until Lesson 4
Lesson-specific adjustments
Standing adjustments are on the linked Personalised Adjustments document and are not repeated here.
- Spinner templates supplied pre-divided into 4, 6 or 8 equal sections. No student draws their own sections — unequal sections would make the probability wrong and the error would be invisible to them.
- Students may observe the floor circle from a seat and handle the spinner cards afterwards. Contributing to the ordering discussion is what matters, not sitting on the floor.
- Tally chart pre-drawn with twenty boxes so the recording is ticking, not constructing.
- Pairs assigned by the teacher, with the spinner-holder and the recorder named before spinning starts.
Misconceptions to expect today
"More sections means a better chance"
Students read a bigger denominator as a bigger probability. It shows up in the Launch ordering — the 8-section spinners get grouped together regardless of how many sections are shaded.
Response. Two spinners side by side: 8 sections with 1 shaded, and 4 sections with 3 shaded. Which would you rather spin? The 8-section one has more sections and a much worse chance.
1/4 and 2/8 treated as different probabilities
Predictable, and it is the point of the lesson rather than a failure. Students see two different-looking spinners and assume two different chances.
Response. Physical, not symbolic. Lay the 8-section spinner over the 4-section one and let them see the two shaded eighths cover the one shaded quarter exactly.
"The probability is 6 out of 20 because that's what I spun"
The observed result is taken as the probability. This is the misconception the whole cycle is built to address, and today it should be named but not resolved.
Response. Ask what they would have said if they had spun it only twice. Then leave it. Lesson 4 does this properly.
Counting the shaded sections but not all the sections
A student writes 3 as the probability, or 3/5 when there are 8 sections and 5 are unshaded — comparing shaded to unshaded rather than shaded to total.
Response. "Show me the 8 on your spinner." Have them touch each section as they count. The denominator has to be pointed at, not recalled.
What comes next
Lesson 3 — What might the deck be? An open task that runs today's thinking backwards: given the probabilities, what could the sample space be? It is a withdrawal-safe consolidation lesson, so it introduces no new content and carries a five-minute rejoin note.